I know that a measurable space is a tuple $(X,\Sigma)$, where $X$ is a set and $\Sigma$ is a $\sigma$-algebra of (measurable) subsets of $X$. A measure space, in contrast is a triple $(X,\Sigma,\mu)$, where $X$ is a set, $\Sigma$ is a $\sigma$-algebra of (measurable) subsets of $X$, and $\mu$ is a measure.
The obvious main difference is that a measurable space does not require a specific measure. Does this mean that measurability as a concept is completely independent of any specific measure? Are $\sigma$-algebras always measurable by any measure, or are they considered measurable if at least one valid measure exists? If measurability is a concept independent of specific measures, why is there a correspondence between, for example, Borel $\sigma$-algebras and the Borel measure, or Lebesque $\sigma$-algebras and the Lebesgue measure?
Perhaps someone can clarify and point me to some resources that make this clear. Thank you!